Problem 23 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(21-26,\) find \(\lim _{x \rightarrow \infty} y\) and \(\lim _{x \rightarrow-\infty} y\). $$y=\frac{\cos (1 / x)}{1+(1 / x)}$$

Step-by-Step Solution

Answer

The limits of \(y\) as \(x \rightarrow \infty\) and \(x \rightarrow -\infty\) are both 1.

Step 1: Analyze the limits approaching infinity

The core of the issue is, when \(x\) tends to infinity (\(x \rightarrow \infty \)) or negative infinity (\(x \rightarrow -\infty \)), the term \(1 / x\) will trend towards 0, because any finite number divided by an infinite one effectively becomes 0. Thus, both the numerator and the denominator within the function will look like this: Numerator: \(\cos (1 / x) \rightarrow \cos (0) = 1\)Denominator: \(1+(1 / x) \rightarrow 1 + 0 = 1\)Therefore, as \(x \rightarrow \infty\) or \(x \rightarrow -\infty\), the entire function will look like this: \(y=\frac{1}{1} = 1\)

Step 2: Compute the limits

From the above step, we can see that both \(\lim _{x \rightarrow \infty} y\) and \(\lim _{x \rightarrow -\infty} y\) should equal to 1. This is because as \(x\) approaches positive or negative infinity, the value of \(y\) trends towards 1. Therefore, we can simplify by saying that the limits of \(y\) as \(x\) tends to infinity (both positive and negative), is 1.

Key Concepts

Limits Approaching Infinity

When we discuss limits approaching infinity, we are considering the behavior of a function as the independent variable, typically denoted as 'x', grows without bound either positively (\( x \rightarrow \text{infinity} \) or negatively (\( x \rightarrow -\text{infinity} \) . This is essential in calculus as it gives us insight into the end behavior of functions, regardless of the complexities involved within their expressions.

Understanding limits at infinity is like watching an arrow in flight; as it goes further and further, its position relative to the start fades in significance. Similarly, as 'x' increases or decreases without restriction, certain terms in the function's formula become negligible, especially terms that involve fractions with 'x' in the denominator. As such, evaluating a limit at infinity often involves simplifying the function, leaving only the most dominant terms that dictate the function's trajectory as 'x' moves towards infinity.

Limit of a Function

The concept of the limit of a function is foundational in calculus, representing the value that a function approaches as the input approaches a certain point. We often denote it symbolically as \( \lim_{x \rightarrow c} f(x) \) where 'c' can be any number, infinity, or negative infinity. Limits help describe the behavior of functions at points where they might not be explicitly defined or where they exhibit interesting behavior, such as discontinuities.

In the provided exercise, the limit of the function \( y=\frac{\cos (1 / x)}{1+(1 / x)} \) as 'x' approaches infinity is being explored. The core process resolves around identifying which parts of 'y' remain significant as 'x' escalates towards infinity — quite like separating the wheat from the chaff. The goal is a clear understanding of the function's end behavior without getting tangled up in the mathematically inconsequential details.

Cosine Function Behavior

The cosine function behavior is of particular interest in trigonometry and calculus. The standard cosine function, denoted as \( \cos(x) \) , oscillates between -1 and 1 for all real number inputs. It depicts wave-like behavior as 'x' varies, with important properties such as periodicity and even symmetry.

However, in the context of limits, particularly those approaching infinity, what matters is the behavior as the argument of the cosine function gets infinitesimally small. The function \( \cos(x) \) approaches 1 as its argument goes to 0, which is an application of the limit concept: \( \lim_{x \rightarrow 0} \cos(x) = 1 \) . This principle plays a crucial role in the exercise at hand, wherein the \( \cos(1/x) \) tends toward \( \cos(0) \) as 'x' approaches infinity, effectively simplifying the evaluation of the original function's limit.